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International Journal of Engineering, Science and Mathematics
Vol. 5 Issue 1, March 2016, ISSN: 2320-0294, Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com
Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-
Gage as well as in Caell’s Diretories of Puli
shing Opportunities, U.S.A
230
International Journal of Engineering, Science and Mathematics
http://www.ijesm.co.in, Email: ijesmj@gmail.com
Wave Propagation through Loosely Bonded Solid/Solid Interface
Vinod Kaliraman Department of Mathematics, Chaudhary Devi Lal University, Sirsa-125055, INDIA
Abstract
In this paper, solution of the governing equations of micropolar elastic solid and fluid saturated incompressible porous solid is employed to study the reflection and transmission phenomenon at a loosely bonded interface between micropolar elastic solid half space and fluid saturated porous half space. P-wave or SV-wave is considered to be incident on the plane interface through fluid saturated porous solid half space. The amplitude ratios of various reflected and transmitted waves are derived and computed numerically for a specific model for different values of bonding parameter. The results thus obtained are depicted graphically with angle of incidence of incident wave. It is found that these amplitude ratios depend on angle of incidence of the incident wave and material properties of the medium. Effect of bonding parameter, fluid filled in the pores of fluid saturated porous medium on the amplitude ratios is shown.
Keywords:
Porous solid, micropolar elastic solid, reflection, transmission, longitudinal wave, transverse wave, amplitude ratios, empty porous solid, loosely bonded interface.
1.
Introduction
Most of natural and man-made materials, including engineering, geological and biological media, possess a microstructure. The ordinary classical theory of elasticity fails to describe the microstructure of the material. To tackle this problem, Suhubi and Eringen (1964), Eringen and Suhubi (1964) developed a theory in which they considered the microstructure of the material and they showed that the motion in a granular structure material is characterized not by a displacement vector but also by a rotation vector. Gautheir (1982) found aluminum-epoxy composite to be a micropolar material. Many problems of waves and vibrations have been discussed in micropolar elastic solid by several researchers. Some of them are Parfitt and Eringen (1969), Tomar and Gogna (1992), Tomar and Kumar (1995), Singh and Kumar (2007), Kumar and Barak (2007) etc. In the problems of wave propagation at the interface between two elastic half spaces, the contact between them is normally assumed to be welded. However, in certain situations, there are reasons for expecting that bonding is not complete. Murty (1976) discussed a theoretical model for reflection, transmission, and attenuation of elastic waves through a loosely bonded interface between two elastic solid half spaces by assuming that the interface behaves like a dislocation which preserves the continuity of stresses allowing a finite amount of slip. A similar situation occurs at the two different poroelastic solids, as the liquid present in the porous skeleton may cause the two media to be loosely bonded. Vashisth and Gogna (1993), Kumar and Singh (1997) etc. discussed the problems of reflection and transmission at the loosely bonded interface between two half spaces. The mechanical behaviour of fluid saturated porous material when the material contains liquid filled pores with help of classical theory is inadequate. Due to complicated structures of pores and different motions of solid and liquid phases, it is very complex and difficult to describe the mechanical behaviour of a fluid saturated porous medium. So many researchers tried to overcome this difficulty from time to time. Bowen (1980) and de Boer and Ehlers (1990a, 1990b) developed an interesting theory for porous medium having all constituents to be incompressible. There are sufficient reasons for considering the fluid saturated porous
International Journal of Engineering, Science and Mathematics
Vol. 5, Issue 1, March 2016, ISSN: 2320-0294, Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com
Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-
Gage as well as in Caell’s Diretories of Pulishing Opportunities, U.S.A
231
International Journal of Engineering, Science and Mathematics
http://www.ijesm.co.in, Email: ijesmj@gmail.com
constituents as incompressible. Therefore, the assumption of incompressible constituents meet the properties appearing the in many branches of engineering and avoids the introduction of many complicated material parameters as considered in the Biot theory
because ψiot’s theory was based on the assumption of compressible constituents.
Based on the theory given by de Boer and Ehlers (1990a, 1990b), many researchers like de Boer and Didwania (2004), de Boer and Liu(1994,1995), Kumar and Hundal (2003), Tajuddin and Hussaini (2006), Kumar et.al.(2011), Kumari (2014) etc. studied some problems of wave propagation in fluid saturated porous media. Using the theory of de Boer and Ehlers (1990) for fluid saturated porous medium and Eringen (1968) theory for micropolar elastic solid, the reflection and transmission of longitudinal wave (P-wave) or transverse wave (SV-wave) at a loosely bonded interface between micropolar elastic solid half space and fluid saturated porous solid half space is discussed. Amplitudes ratios for various reflected and transmitted waves are computed for a particular model and depicted graphically and discussed accordingly. The model considered is assumed to exist in the oceanic
crust part of the earth and the propagation of wave through such a model will be of great use in the fields related to earth sciences.
2.
Formulation of the problem
Consider a two dimensional problem by taking the z-axis pointing into the lower half-space and the plane interface z=0 separating the fluid saturated porous half space
M
[μ>]
and micropolar elastic solid half space
M
[z<0]. A longitudinal wave or transverse wave propagates through the medium
M
and incident at the plane z=0 and making an angle
with normal to the surface. Corresponding to incident longitudinal or transverse wave, we get two reflected waves in the medium
M
and three transmitted waves in medium
M
. See fig.1
Fig.1 Geometry of the problem.
3.
Basic equations and constitutive relatio 3.1. For medium
(Micropolar elastic solid half space)
The equation of motion in micropolar elastic medium are given by Eringen (1968) as
c
c
∇
ϕ∂
ϕ∂t
c
c
∇
U⃗c
∇×
Φ
⃗∂
U⃗∂t
c
∇
Φ
⃗
∇×U⃗∂
Φ
⃗∂t
Where
International Journal of Engineering, Science and Mathematics
Vol. 5, Issue 1, March 2016, ISSN: 2320-0294, Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com
Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-
Gage as well as in Caell’s Diretories of Pulishing Opportunities, U.S.A
232
International Journal of Engineering, Science and Mathematics
http://www.ijesm.co.in, Email: ijesmj@gmail.com
c
+
; c
; c
; c
γ
;
Parfitt and Eringen (1969) have shown that equation (1) corresponds to longitudinal wave propagating with velocity
v
, given by
v
c
c
and equations (2)-(3) are coupled equations in vector potentials
U⃗
and
Φ
⃗
and these correspond to coupled transverse and micro-rotation waves. If
ω
0
>,
there exist two sets of coupled-wave propagating with velocities 1/
and 1/
; where
B√ B
C,
B√ B
C,
where
Bqp
c
c
c
; Cc
q
c
c
; p ; q
Considering a two dimensional problem by taking the following components of displacement and micro rotation as
u⃗u,,w,
Φ
⃗,
Φ
,
where
u∂ϕ∂κ∂∂μ w∂ϕ∂μ∂∂κ
and components of stresses are as under
t
∂
ϕ∂μ
∂
ϕ∂κ
∂
∂κ∂μ
t
∂
ϕ∂κ∂μ∂
∂μ
∂
∂κ
Φ
m
Φ
3.2. For medium
(Fluid saturated incompressible porous solid half space)
Following de Boer and Ehlers (1990b), the governing equations in a fluid-saturated incompressible porous medium are
div
η
η
.
div
η
grad p
ρ
,
div
η
grad p
ρ
,
where
and
iS,F
denote the velocities and accelerations, respectively of solid (S) and fluid (F) phases of the porous aggregate and p is the effective pore pressure of the incompressible pore fluid.
ρ
and
ρ
are the densities of the solid and fluid phases respectively and
b
is the body force per unit volume.
and
are the effective stress in the solid and fluid phases respectively,
is the effective quantity of momentum supply and
η
and
η
are the volume fractions satisfying
η
η
.
If
and
are the displacement vectors for solid and fluid phases, then
κ
;
;
;
The constitutive equations for linear isotropic, elastic incompressible porous medium are given by de Boer, Ehlers and Liu (1993) as
E
.,
,
,
International Journal of Engineering, Science and Mathematics
Vol. 5, Issue 1, March 2016, ISSN: 2320-0294, Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com
Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-
Gage as well as in Caell’s Diretories of Pulishing Opportunities, U.S.A
233
International Journal of Engineering, Science and Mathematics
http://www.ijesm.co.in, Email: ijesmj@gmail.com
where
and
are the macroscopic Lame’s parameters of the porous solid and
is the linearized Langrangian strain tensor defined as
grad
grad
,
In the case of isotropic permeability, the tensor
describing the coupled interaction between the solid and fluid is given by de Boer and Ehlers (1990b) as
η
γ
K
,
where
γ
is the specific weight of the fluid and
K
is the Darcy’s permeability coefficient
of the porous medium. Making the use of (16) in equations (12)-(1 4), and with the help of (17)-(20), we obtain
div
η
η
,
(
)grad div
div grad
η
grad p
ρ
S
,
η
grad p
ρ
S
.
For the two dimensional problem, we assume the displacement vector
iF,S
as
(u
,,w
)
where
iF,S.
Equations (22) - (24) with the help of eq. (25) in absence of body forces take the form
∂
u
∂κ∂t∂
w
∂μ∂t
∂
u
∂κ∂t∂
w
∂μ∂t,
∂p∂κρ
∂
u
∂t
S
∂u
∂t∂u
∂t,
∂p∂μρ
∂
w
∂t
S
∂w
∂t∂w
∂t,
∂
∂κ
∇
u
∂p∂κρ
∂
u
∂t
S
∂u
∂t∂u
∂t,
∂
∂μ
∇
w
∂p∂μρ
∂
w
∂t
S
∂w
∂t∂w
∂t,
where
∂u
∂κ∂w
∂μ
and
∇
∂
∂κ
∂
∂μ
Also,
t
and
t
the normal and tangential stresses in the solid phase are as under
t
∂u
∂κ∂w
∂μ
∂w
∂μ
t
∂u
∂μ∂w
∂κ
The displacement components
u
and
w
are related to the dimensional potential
ϕ
and
as
u
∂ϕ
∂κ∂
∂μ ; w
∂ϕ
∂μ∂
∂κ ; jS,F.
Using equation (35) in equations (26)-(30), we obtain the following equations determining
ϕ
, ϕ
,
,
and p
as:
∇
ϕ
C
∂
ϕ
∂t
S
∂ϕ
∂t,
International Journal of Engineering, Science and Mathematics
Vol. 5, Issue 1, March 2016, ISSN: 2320-0294, Impact Factor: 6.765 Journal Homepage: http://www.ijesm.co.in, Email: ijesmj@gmail.com
Double-Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at: Ulrich's Periodicals Directory ©, U.S.A., Open J-
Gage as well as in Caell’s Diretories of Pulishing Opportunities, U.S.A
234
International Journal of Engineering, Science and Mathematics
http://www.ijesm.co.in, Email: ijesmj@gmail.com
ϕ
ϕ
∇
ρ
∂
∂t
S
∂
∂t∂
∂t
ρ
∂
∂t
S
∂
∂t∂
∂t
p
ρ
∂
ϕ
∂t
S
∂ϕ
∂t
where
C
ρ
ρ
Assuming the solution of the system of equations (36) - (40) in the form
ϕ
,ϕ
,
,
,p(ϕ
,ϕ
,
,
,p
)eκpit
where
is the complex circular frequency. Making the use of (42) in equations (36)-(40), we obtain
∇
C
iS
ϕ
[
∇
ρ
iS
]
iS
[
ρ
iS
]
iS
p
ρ
ϕ
iS
ϕ
ϕ
ϕ
Equation (43) corresponds to longitudinal wave propagating with velocity
v
, given by
v
G
where
G
C
iS
From equation (44) and (45), we obtain
∇
v
Equation (50) corresponds to transverse wave propagating with velocity
v
, given by
v
/G
Where
G
ω
(−
ω
+ω
)
In medium
ϕB
eκp{i
(κ sin
μ cos
)i
t},
B
eκp{i
(κ sin
μ cos
)i
t}B
eκp{i
(κ sin
μ cos
)i
t},

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